Finite Group Databasebeta

← C3×#79239 #11868649

Subgroups — C3×#79239 #11868649

Order 1152.

939 conjugacy classes of subgroups.

The subgroup lattice has 597 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5517C211yesmaximal, characteristic, complemented
2C2×C3×#893C211yesmaximal, characteristic
3C32×#428C211yesmaximal, characteristic, complemented
4C3×#5131C211yesmaximal, characteristic
5C3×#5118C211yesmaximal, characteristic, complemented
6C3×#5001C211yesmaximal, characteristic
7C3×#4964C211yesmaximal, characteristic, complemented
8C3×#4891C212yesmaximal, complemented
9C3×#4890C212yesmaximal, complemented
10C3×#4871C212yesmaximal, complemented
11C3×#4870C212yesmaximal, complemented
12C3×#3077—31nomaximal
13#79239C311yesmaximal, characteristic, direct product factor
14C22×C3×D12C2211yescharacteristic
15C22×C3×C4×D3—21no—
16C2×C32×M16C2211yescharacteristic
17C32×#117C2211yescharacteristic
18C2×C32×C4⋊C4C2211yescharacteristic
19C2×C3×(C2×D3)⋊C4—21no—
20C3×#926C2211yescharacteristic
21C2×C3×C4×Dic3C2211yescharacteristic
22C2×C3×#208C2211yescharacteristic
23C3×#893—21no—
24C3×#893—21no—
25C3×#893C2212yes—
26C3×#893C2212yes—
27C3×#890C2211yescharacteristic, complemented
28C3×#890C2211yescharacteristic, complemented
29C3×#888C2211yescharacteristic, complemented
30C3×#888C2211yescharacteristic, complemented
31C3×#878—21no—
32C3×#878—21no—
33C3×C4×Dic6—21no—
34C3×C4×Dic6—21no—
35C32×#107C2212yescomplemented
36C32×#106C2212yescomplemented
37C3×#821C2212yes—
38C3×#821C2212yes—
39C3×Dic3⋊C8C2212yescomplemented
40C3×Dic3⋊C8C2212yescomplemented
41C3×#809C2212yes—
42C3×#809C2212yes—
43C3×#808C2212yescomplemented
44C3×#807C2212yescomplemented
45C3×#533—31no—
46C2×C3×C4×D4—31no—
47#5517C2×C311yescharacteristic, complemented
48C2×#893C2×C311yescharacteristic
49C3×#478—32no—
50C3×#478—32no—
51C3×#476—32no—
52C3×#474—32no—
53C3×#428—21no—
54C3×#428—31no—
55C3×#428D311yescharacteristic, complemented
56C3×#428C2×C311yescharacteristic, complemented
57C3×#423—31no—
58C3×#418—31no—
59C3×#418—31no—
60#5131C2×C311yescharacteristic
61#5118C2×C311yescharacteristic, complemented
62#5001C2×C311yescharacteristic
63#4964C2×C311yescharacteristic, complemented
64#4891C2×C312yescomplemented
65#4890C2×C312yescomplemented
66#4871C2×C312yescomplemented
67#4870C2×C312yescomplemented
68C23×C3×D3—21no—
69C22×C32×C4—21no—
70C22×C32×C4C2311yescharacteristic
71C22×C3×Dic3D411yescharacteristic
72C22×C3×Dic3D411yescharacteristic
73C2×C3×D12—21no—
74C2×C3×D12—22no—
75C2×C3×D12Q814yescomplemented
76C2×C3×D12C2312yes—
77C2×C3×C4×D3—22no—
78C2×C3×C4×D3—22no—
79C2×C3×C4×D3—44no—
80C2×C3×Dic6—21no—
81C2×C3×Dic6—21no—
82C32×M16—22no—
83C2×C32×C8C2312yes—
84C32×C4⋊C4—21no—
85C32×C4⋊C4C2311yescharacteristic
86C32×C4⋊C4C2311yescharacteristic
87C32×C4⋊C4C2312yes—
88C32×C22⋊C4—21no—
89C32×C42—21no—
90C3×#217—21no—
91C3×(C2×D3)⋊C4—22no—
92C3×(C2×D3)⋊C4—22no—
93C3×C4⋊Dic3—21no—
94C3×C4⋊Dic3—21no—
95C3×C4⋊Dic3C2311yescharacteristic
96C3×C4⋊Dic3C2311yescharacteristic
97C3×Dic3⋊C4—21no—
98C3×Dic3⋊C4—21no—
99C3×Dic3⋊C4—21no—
100C3×Dic3⋊C4—21no—
101C3×C4×Dic3—21no—
102C3×C4×Dic3D411yescharacteristic
103C3×C4×Dic3D411yescharacteristic
104C3×C4×Dic3C2311yescharacteristic
105C3×C4×Dic3C2311yescharacteristic
106C3×#208—22no—
107C2×C3×C3⋊C8C2312yes—
108#3077—31noSylow
109C22×C3×D4—31no—
110C23×C3×C4—61no—
111C22×D12C22×C311yescharacteristic
112C22×C4×D3—21no—
113C2×C3×M16—21no—
114C2×C3×M16—31no—
115C2×C3×M16C2×D311yescharacteristic
116C2×C3×M16C22×C311yescharacteristic
117C3×C4⋊Q8—31no—
118C3×C4⋊Q8—31no—
119C3×#125—31no—
120C3×#125—31no—
121C3×C22⋊Q8—61no—
122C3×C22⋊Q8—61no—
123C3×C4×Q8—61no—
124C3×C4×Q8—61no—
125C3×C4×D4—32no—
126C3×C4×D4—32no—
127C3×C4×D4—61no—
128C3×C4×D4—61no—
129C3×#117—21no—
130C3×#117—31no—
131C3×#117C2×D311yescharacteristic
132C3×#117C22×C311yescharacteristic
133C2×C3×C4⋊C4—21no—
134C2×C3×C4⋊C4C2×D311yescharacteristic
135C2×C3×C4⋊C4C22×C311yescharacteristic
136C2×C3×C22⋊C4—61no—
137C2×C3×C42—31no—
138C2×(C2×D3)⋊C4—21no—
139#926C22×C311yescharacteristic
140C2×C4×Dic3C22×C311yescharacteristic
141C2×#208C22×C311yescharacteristic
142#893—21no—
143#893—21no—
144#893C22×C312yes—
145#893C22×C312yes—
146#890C22×C311yescharacteristic, complemented
147#890C22×C311yescharacteristic, complemented
148#888C22×C311yescharacteristic, complemented
149#888C22×C311yescharacteristic, complemented
150#878—21no—
151#878—21no—
152C4×Dic6—21no—
153C4×Dic6—21no—
154C3×#107—22no—
155C3×#107—32no—
156C3×#107C2×D312yescomplemented
157C3×#107C22×C312yescomplemented
158C3×#106—22no—
159C3×#106—32no—
160C3×#106C2×D312yescomplemented
161C3×#106C22×C312yescomplemented
162C3×C4⋊C8—32no—
163C3×C4⋊C8—32no—
164C3×#102—32no—
165C3×#102—32no—
166C3×#102—32no—
167C3×#102—32no—
168#821C22×C312yes—
169#821C22×C312yes—
170Dic3⋊C8C22×C312yescomplemented
171Dic3⋊C8C22×C312yescomplemented
172#809C22×C312yes—
173#809C22×C312yes—
174#808C22×C312yescomplemented
175#807C22×C312yescomplemented
176C23×C32C2×D411yescharacteristic
177C22×C3×D3—24no—
178C22×C3×D3—22no—
179C22×C3×D3—42no—
180C22×C3×D3—42no—
181C2×C32×C4—21no—
182C2×C32×C4—21no—
183C2×C32×C4—22no—
184C2×C32×C4—42no—
185C2×C32×C4C2×D411yescharacteristic
186C2×C32×C4C2×Q812yes—
187C2×C32×C4#4112yes—
188C2×C32×C4C2411yescharacteristic
189C2×C3×Dic3—22no—
190C2×C3×Dic3—21no—
191C2×C3×Dic3—21no—
192C2×C3×Dic3—41no—
193C2×C3×Dic3C2×D411yescharacteristic
194C2×C3×Dic3C2×D411yescharacteristic
195C2×C3×Dic3C2×D411yescharacteristic
196C2×C3×Dic3C2×D411yescharacteristic
197C3×D12—24no—
198C3×D12—22no—
199C3×D12C2×Q814yes—
200C3×C4×D3—44no—
201C3×C4×D3—44no—
202C3×Dic6—42no—
203C3×Dic6—42no—
204C32×C8—22no—
205C3×C3⋊C8—22no—
206#533—31no—
207C2×C4×D4—31no—
208#478—32no—
209#478—32no—
210#476—32no—
211#474—32no—
212#428—31no—
213#428C3×D311yescharacteristic, complemented
214#423—31no—
215#418—31no—
216#418—31no—
217C24×C3—61no—
218C23×D3—21no—
219C2×C3×Q8—61no—
220C2×C3×Q8—61no—
221C2×C3×D4—34no—
222C2×C3×D4—32no—
223C2×C3×D4—62no—
224C2×C3×D4—61no—
225C22×C3×C4—21no—
226C22×C3×C4—21no—
227C22×C3×C4—21no—
228C22×C3×C4—31no—
229C22×C3×C4—31no—
230C22×C3×C4—41no—
231C22×C3×C4—62no—
232C22×C3×C4—62no—
233C22×C3×C4—124no—
234C22×C3×C4C22×D311yescharacteristic
235C22×C3×C4C23×C311yescharacteristic
236C22×Dic3C3×D411yescharacteristic
237C22×Dic3C3×D411yescharacteristic
238C2×D12—21no—
239C2×D12—22no—
240C2×D12C3×Q814yescomplemented
241C2×D12C23×C312yes—
242C2×C4×D3—22no—
243C2×C4×D3—22no—
244C2×C4×D3—44no—
245C2×Dic6—21no—
246C2×Dic6—21no—
247C3×M16—24no—
248C3×M16—22no—
249C3×M16—22no—
250C3×M16—62no—
251C2×C3×C8—22no—
252C2×C3×C8—32no—
253C2×C3×C8C22×D312yes—
254C2×C3×C8C23×C312yes—
255C3×C4⋊C4—21no—
256C3×C4⋊C4—21no—
257C3×C4⋊C4—21no—
258C3×C4⋊C4—22no—
259C3×C4⋊C4—21no—
260C3×C4⋊C4—31no—
261C3×C4⋊C4—31no—
262C3×C4⋊C4—41no—
263C3×C4⋊C4—61no—
264C3×C4⋊C4—61no—
265C3×C4⋊C4—61no—
266C3×C4⋊C4—61no—
267C3×C4⋊C4—61no—
268C3×C4⋊C4—61no—
269C3×C4⋊C4C22×D311yescharacteristic
270C3×C4⋊C4C22×D311yescharacteristic
271C3×C4⋊C4C22×D312yes—
272C3×C4⋊C4C23×C312yes—
273C3×C4⋊C4C23×C311yescharacteristic
274C3×C4⋊C4C23×C311yescharacteristic
275C3×C22⋊C4—21no—
276C3×C22⋊C4—21no—
277C3×C22⋊C4—41no—
278C3×C22⋊C4—62no—
279C3×C22⋊C4—62no—
280C3×C22⋊C4—61no—
281C3×C42—21no—
282C3×C42—21no—
283C3×C42—31no—
284C3×C42—31no—
285C3×C42—31no—
286C3×C42—31no—
287C3×C42—41no—
288C3×C42—61no—
289#217—21no—
290(C2×D3)⋊C4—22no—
291(C2×D3)⋊C4—22no—
292C4⋊Dic3—21no—
293C4⋊Dic3—21no—
294C4⋊Dic3C23×C311yescharacteristic
295C4⋊Dic3C23×C311yescharacteristic
296Dic3⋊C4—21no—
297Dic3⋊C4—21no—
298Dic3⋊C4—21no—
299Dic3⋊C4—21no—
300C4×Dic3—21no—
301C4×Dic3C3×D411yescharacteristic
302C4×Dic3C3×D411yescharacteristic
303C4×Dic3C23×C311yescharacteristic
304C4×Dic3C23×C311yescharacteristic
305#208—22no—
306C2×C3⋊C8C23×C312yes—
307C22×C32—21no—
308C22×C32—21no—
309C22×C32C22⋊Q812yes—
310C22×C32C22×D411yescharacteristic
311C2×C3×D3—24no—
312C2×C3×D3—48no—
313C2×C3×D3—42no—
314C2×C3×D3—42no—
315C32×C4—41no—
316C32×C4—42no—
317C32×C4—41no—
318C32×C4C22⋊Q812yes—
319C32×C4C22×Q811yescharacteristic
320C32×C4C2×#4111yescharacteristic
321C3×Dic3—21no—
322C3×Dic3—21no—
323C3×Dic3—21no—
324C3×Dic3—41no—
325C3×Dic3—41no—
326C3×Dic3AGL(1,ℤ/8)12yes—
327C22×D4—31no—
328C23×C4—61no—
329C2×M16—31no—
330C2×M16C2×C3×D311yescharacteristic
331C4⋊Q8—31no—
332C4⋊Q8—31no—
333#125—31no—
334#125—31no—
335C22⋊Q8—61no—
336C22⋊Q8—61no—
337C4×Q8—61no—
338C4×Q8—61no—
339C4×D4—32no—
340C4×D4—32no—
341C4×D4—61no—
342C4×D4—61no—
343#117—31no—
344#117C2×C3×D311yescharacteristic
345C2×C4⋊C4C2×C3×D311yescharacteristic
346C2×C22⋊C4—61no—
347C2×C42—31no—
348#107—32no—
349#107C2×C3×D312yescomplemented
350#106—32no—
351#106C2×C3×D312yescomplemented
352C4⋊C8—32no—
353C4⋊C8—32no—
354#102—32no—
355#102—32no—
356#102—32no—
357#102—32no—
358C23×C3—21no—
359C23×C3—64no—
360C23×C3—62no—
361C23×C3—122no—
362C23×C3—122no—
363C23×C3D3×D411yescharacteristic
364C23×C3C2×C3×D411yescharacteristic
365C22×D3—24no—
366C22×D3—22no—
367C22×D3—42no—
368C22×D3—42no—
369C3×Q8—122no—
370C3×Q8—122no—
371C3×D4—34no—
372C3×D4—64no—
373C3×D4—62no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—22no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—22no—
380C2×C3×C4—21no—
381C2×C3×C4—22no—
382C2×C3×C4—21no—
383C2×C3×C4—22no—
384C2×C3×C4—31no—
385C2×C3×C4—31no—
386C2×C3×C4—31no—
387C2×C3×C4—31no—
388C2×C3×C4—41no—
389C2×C3×C4—41no—
390C2×C3×C4—42no—
391C2×C3×C4—42no—
392C2×C3×C4—42no—
393C2×C3×C4—62no—
394C2×C3×C4—61no—
395C2×C3×C4—61no—
396C2×C3×C4—82no—
397C2×C3×C4—124no—
398C2×C3×C4—124no—
399C2×C3×C4—121no—
400C2×C3×C4#23512yes—
401C2×C3×C4D3×D411yescharacteristic
402C2×C3×C4D3×Q812yes—
403C2×C3×C4C2×C3×D411yescharacteristic
404C2×C3×C4C2×C3×Q812yes—
405C2×C3×C4C3×#4112yes—
406C2×C3×C4C23×D311yescharacteristic
407C2×C3×C4C24×C311yescharacteristic
408C2×Dic3—22no—
409C2×Dic3—21no—
410C2×Dic3—21no—
411C2×Dic3—41no—
412C2×Dic3C2×C3×D411yescharacteristic
413C2×Dic3C2×C3×D411yescharacteristic
414C2×Dic3C2×C3×D411yescharacteristic
415C2×Dic3C2×C3×D411yescharacteristic
416D12—24no—
417D12—22no—
418D12C2×C3×Q814yes—
419C4×D3—44no—
420C4×D3—44no—
421Dic6—42no—
422Dic6—42no—
423C3×C8—24no—
424C3×C8—22no—
425C3×C8—22no—
426C3×C8—62no—
427C3⋊C8—22no—
428C2×C32—22no—
429C2×C32C2×C22⋊Q811yescharacteristic
430C2×C32C2×AGL(1,ℤ/8)11yescharacteristic
431C2×C32#57511yescharacteristic
432C3×D3—44no—
433C24—61no—
434C2×Q8—61no—
435C2×Q8—61no—
436C2×D4—32no—
437C2×D4—34no—
438C2×D4—62no—
439C2×D4—61no—
440C22×C4—21no—
441C22×C4—31no—
442C22×C4—31no—
443C22×C4—62no—
444C22×C4—62no—
445C22×C4—124no—
446C22×C4C22×C3×D311yescharacteristic
447M16—22no—
448M16—62no—
449C2×C8—32no—
450C2×C8C22×C3×D312yes—
451C4⋊C4—21no—
452C4⋊C4—31no—
453C4⋊C4—31no—
454C4⋊C4—61no—
455C4⋊C4—61no—
456C4⋊C4—61no—
457C4⋊C4—61no—
458C4⋊C4—61no—
459C4⋊C4—61no—
460C4⋊C4C22×C3×D311yescharacteristic
461C4⋊C4C22×C3×D311yescharacteristic
462C4⋊C4C22×C3×D312yes—
463C22⋊C4—21no—
464C22⋊C4—62no—
465C22⋊C4—62no—
466C22⋊C4—61no—
467C42—21no—
468C42—31no—
469C42—31no—
470C42—31no—
471C42—31no—
472C42—61no—
473C22×C3—21no—
474C22×C3—21no—
475C22×C3—21no—
476C22×C3—21no—
477C22×C3—22no—
478C22×C3—21no—
479C22×C3—41no—
480C22×C3—41no—
481C22×C3—64no—
482C22×C3—128no—
483C22×C3—122no—
484C22×C3—122no—
485C22×C3#89612yes—
486C22×C3C3×C22⋊Q812yes—
487C22×C3C2×D3×D411yescharacteristic
488C22×C3C22×C3×D411yescharacteristic
489C2×D3—24no—
490C2×D3—48no—
491C2×D3—42no—
492C2×D3—42no—
493C3×C4—22no—
494C3×C4—21no—
495C3×C4—21no—
496C3×C4—32no—
497C3×C4—41no—
498C3×C4—42no—
499C3×C4—41no—
500C3×C4—42no—
501C3×C4—41no—
502C3×C4—41no—
503C3×C4—61no—
504C3×C4—61no—
505C3×C4—61no—
506C3×C4—82no—
507C3×C4—81no—
508C3×C4—81no—
509C3×C4—121no—
510C3×C4—121no—
511C3×C4#89612yes—
512C3×C4C3×C22⋊Q812yes—
513C3×C4C2×#23511yescharacteristic
514C3×C4C2×D3×Q811yescharacteristic
515C3×C4C22×C3×Q811yescharacteristic
516C3×C4C2×C3×#4111yescharacteristic
517Dic3—21no—
518Dic3—21no—
519Dic3—21no—
520Dic3—41no—
521Dic3—41no—
522Dic3C3×AGL(1,ℤ/8)12yes—
523C32#307711yescharacteristic, complemented, Sylow
524C23—64no—
525C23—62no—
526C23—122no—
527C23—122no—
528C23C3×D3×D411yescharacteristic
529Q8—122no—
530Q8—122no—
531D4—34no—
532D4—64no—
533D4—62no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—31no—
538C2×C4—31no—
539C2×C4—31no—
540C2×C4—31no—
541C2×C4—42no—
542C2×C4—62no—
543C2×C4—61no—
544C2×C4—61no—
545C2×C4—124no—
546C2×C4—124no—
547C2×C4—121no—
548C2×C4C3×#23512yes—
549C2×C4C3×D3×D411yescharacteristic
550C2×C4C3×D3×Q812yes—
551C2×C4C23×C3×D311yescharacteristic
552C8—22no—
553C8—62no—
554C2×C3—22no—
555C2×C3—22no—
556C2×C3—21no—
557C2×C3—21no—
558C2×C3—21no—
559C2×C3—42no—
560C2×C3—124no—
561C2×C3C2×#89611yescharacteristic
562C2×C3D3×AGL(1,ℤ/8)11yescharacteristic
563C2×C3#577711yescharacteristic
564C2×C3C2×C3×C22⋊Q811yescharacteristic
565C2×C3C2×C3×AGL(1,ℤ/8)11yescharacteristic
566C2×C3C3×#57511yescharacteristic
567D3—44no—
568C22—21no—
569C22—21no—
570C22—64no—
571C22—128no—
572C22—122no—
573C22—122no—
574C22C3×#89612yes—
575C22C2×C3×D3×D411yescharacteristic
576C4—32no—
577C4—42no—
578C4—41no—
579C4—41no—
580C4—61no—
581C4—61no—
582C4—61no—
583C4—121no—
584C4—121no—
585C4C3×#89612yes—
586C4C2×C3×#23511yescharacteristic
587C4C2×C3×D3×Q811yescharacteristic
588C3—21no—
589C3#7923911yescharacteristic, direct product factor
590C3C3×#307711yescharacteristic, complemented
591C2—22no—
592C2—124no—
593C2C2×C3×#89611yescharacteristic
594C2C3×D3×AGL(1,ℤ/8)11yescharacteristic
595C2C3×#577711yescharacteristic
subgroupG/H
centerC22×C3C2×D3×D4
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC32×#428C2
socleC22×C32C22×D4
prime radicalC22×C3×D12C22
hypercenterC3×#428D3
hypocenterC3C3×#3077
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#428

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C3×#5517C2semidirect
3. C32×#428C2semidirect
5. C3×#5118C2semidirect
7. C3×#4964C2semidirect
8. C3×#4891C2semidirect
9. C3×#4890C2semidirect
10. C3×#4871C2semidirect
11. C3×#4870C2semidirect
13. #79239C3direct (N × H)
27. C3×#890C22semidirect
28. C3×#890C22semidirect
29. C3×#888C22semidirect
30. C3×#888C22semidirect
35. C32×#107C22semidirect
36. C32×#106C22semidirect
39. C3×Dic3⋊C8C22semidirect
40. C3×Dic3⋊C8C22semidirect
43. C3×#808C22semidirect
44. C3×#807C22semidirect
47. #5517C2×C3semidirect
55. C3×#428D3semidirect
56. C3×#428C2×C3semidirect
61. #5118C2×C3semidirect
63. #4964C2×C3semidirect
64. #4891C2×C3semidirect
65. #4890C2×C3semidirect
66. #4871C2×C3semidirect
67. #4870C2×C3semidirect
75. C2×C3×D12Q8semidirect
146. #890C22×C3semidirect
147. #890C22×C3semidirect
148. #888C22×C3semidirect
149. #888C22×C3semidirect
156. C3×#107C2×D3semidirect
157. C3×#107C22×C3semidirect
160. C3×#106C2×D3semidirect
161. C3×#106C22×C3semidirect
170. Dic3⋊C8C22×C3semidirect
171. Dic3⋊C8C22×C3semidirect
174. #808C22×C3semidirect
175. #807C22×C3semidirect
213. #428C3×D3semidirect
240. C2×D12C3×Q8semidirect
349. #107C2×C3×D3semidirect
351. #106C2×C3×D3semidirect
523. C32#3077semidirect
589. C3#79239direct (N × H)
590. C3C3×#3077semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C3×#55172
C2×C3×#8932
C32×#4282
C3×#51312
C3×#51182
C3×#50012
C3×#49642
C3×#48912
C3×#48902
C3×#48712
C3×#48702
C3×#30773
#792393